A143941 -id:A143941 - cp's OEIS frontend

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

1, 6, 3, 19, 14, 5, 44, 37, 22, 7, 85, 76, 55, 30, 9, 146, 135, 108, 73, 38, 11, 231, 218, 185, 140, 91, 46, 13, 344, 329, 290, 235, 172, 109, 54, 15, 489, 472, 427, 362, 285, 204, 127, 62, 17, 670, 651, 600, 525, 434, 335, 236, 145, 70, 19, 891, 870, 813

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A100157
Antidiagonal sums:  A071238
row 1,  (1,3,5,7,9,...)**(1,3,5,7,9,...): A005900
row 2,  (1,3,5,7,9,...)**(3,5,7,9,11,...): A143941
row 3,  (1,3,5,7,9,...)**(5,7,9,11,13,...): (2*k^3 + 12*k^2 + k)/6
row 4,  (1,3,5,7,9,...)**(7,9,11,13,15,,...): (2*k^3 + 18*k^2 + k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...6....19...44....85....146
3...14...37...76....135...218
5...22...55...108...185...290
7...30...73...140...235...362
9...38...91...172...285...434

Cf. A213500.

b[n_] := 2 n - 1; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213752 *)
Table[t[n, n], {n, 1, 40}] (* A100157 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A071238 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = 2*n - 1 + 2*x - (2*n - 3)*x^2 and g(x) = (1 - x )^4.

A143940 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!), 1 <= k <= n.

Original entry on oeis.org

3, 6, 4, 9, 8, 4, 12, 12, 8, 4, 15, 16, 12, 8, 4, 18, 20, 16, 12, 8, 4, 21, 24, 20, 16, 12, 8, 4, 24, 28, 24, 20, 16, 12, 8, 4, 27, 32, 28, 24, 20, 16, 12, 8, 4, 30, 36, 32, 28, 24, 20, 16, 12, 8, 4, 33, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4, 36, 44, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4

Offset: 1

Emeric Deutsch, Sep 06 2008

The entries in row n are the coefficients of the Wiener polynomial of a linear chain of n triangles.
Sum of entries in row n = n(2n+1) = A014105(n).
Sum_{k=1..n} k*T(n,k) = the Wiener index of the linear chain of n triangles = A143941(n).

			T(2,1)=6 because the chain of 2 triangles has 6 edges.
Triangle starts:
   3;
   6,  4;
   9,  8,  4;
  12, 12,  8,  4;
  15, 16, 12,  8,  4;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A014105, A143941.

T:=proc(n,k) if n < k then 0 elif k = 1 then 3*n else 4*n-4*k+4 end if end proc: for n to 12 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

T(n,1)=3n; T(n,k) = 4(n-k+1) for k>1.

G.f. = G(q,z) = qz/(3+qz)/((1-qz)*(1-z)^2).

A333411 Number of isomorphism classes which minimize the Wiener index among all triangulations.

Original entry on oeis.org

1, 1, 2, 5, 12, 36, 99, 255, 614, 1532, 3908, 10727, 31242, 96725, 311735

Offset: 4

Stefano Spezia, Mar 20 2020

Éva Czabarka, Trevor Olsen, Stephen Smith and László A. Székely, Minimum Wiener Index of Triangulations and Quadrangulations, arXiv preprint arXiv:2003.03873 [math.CO], 2020.

Cf. A143941, A333412.

A333412 Number of isomorphism classes which minimize the Wiener index among all 4-connected triangulations.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 10, 10, 14, 15, 19, 21, 25, 27, 32, 34, 39, 42

Offset: 4

Stefano Spezia, Mar 20 2020

Éva Czabarka, Trevor Olsen, Stephen Smith and László A. Székely, Minimum Wiener Index of Triangulations and Quadrangulations, arXiv preprint arXiv:2003.03873 [math.CO], 2020.

Cf. A143941, A333411.

cp's OEIS Frontend

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

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A143940 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!), 1 <= k <= n.

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A333411 Number of isomorphism classes which minimize the Wiener index among all triangulations.

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Author

Keywords

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Crossrefs

A333412 Number of isomorphism classes which minimize the Wiener index among all 4-connected triangulations.

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Author

Keywords

Links

Crossrefs